A pair of dice is thrown repeatedly until the total obtained on the first throw is obtained again. What is the expected number of throws necessary?

The answer from book is 12.

Here is my reasoning

There are 11 different totals for a pair of dice (you can't roll a 1)
(2,3,4,5,6,7,8,9,10,11,12)

11 possibilities plus the 1 original throw = 12

Is there a formal way to do this?(Can this be solved by using a specific formula?)

What about the following: let be the number of throws until the first throw appears again, then the expected value of N is
As you can see, it gets a little more complicated than you might have expected...